查看“︁林家翹-钱学森方程”︁的源代码

'''林-钱方程'''（Lin-Tsien equation）是[[林家翹]]-[[钱学森]]和创立的描述可压缩流体中物体跨音速运动的非线性偏微分方程<ref>William F. Ames. [http://books.google.gr/books?id=xwaZP3fxAzIC&dq=non+linear+quations+Tsien+equation&source=gbs_navlinks_s Nonlinear partial differential equations in engineering, Vol. 18] {{Wayback|url=http://books.google.gr/books?id=xwaZP3fxAzIC&dq=non+linear+quations+Tsien+equation&source=gbs_navlinks_s |date=20131214074535 }}. Academic Press, p.&nbsp;173, 1965.</ref><ref>C.C. Lin, E.Reissner,and H.S.Tsien, On Two Dimensional Non-steady Motion of a Slender Body in a Compressible Fluid, 钱学森文集 p513 上海交通大学出版社</ref>

: <math>2u_{tx}+u_xu_{xx}-u_{yy} = 0. \, </math>

==行波解==

利用[[Maple]]中的软件包TWS_solution可得林-钱方程的多种行波解<ref>Graham W. Griffiths  Willian Schiesser, Traveling wave analysis of Partial Differential  Equations p436</ref>
<syntaxhighlight lang="mupad">
u(x,y,t)=TWS_sol := {u(x, y, t) = _C1+(-(1/2)*ln(tanh(_C3+_C4*x+_C5*y+_C6*t)+1)+(1/2)*ln(tanh(_C3+_C4*x+_C5*y+_C6*t)-1))*_C2}
g[2] := {u(x, y, t) = _C1+arctan(1/\sqrt(csc(_C3+_C4*x+_C5*y+_C6*t)^2-1))*_C2}
g[2] := {u(x, y, t) = _C1+(-(1/2)*ln(coth(_C3+_C4*x+_C5*y+_C6*t)+1)+(1/2)*ln(coth(_C3+_C4*x+_C5*y+_C6*t)-1))*_C2}
g[2] := {u(x, y, t) = _C1+arctanh(1/\sqrt(1+csch(_C3+_C4*x+_C5*y+_C6*t)^2))*_C2}
</syntaxhighlight>
找到了一个新的可积（3 + 1）维泛化; 见论文 <ref>A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108 (2018), no. 2, 359-376, https://arxiv.org/abs/1401.2122 {{Wayback|url=https://arxiv.org/abs/1401.2122 |date=20201112013741 }}</ref>的系统（40）

==行波图==
{|
|[[File:Lin-Tsien equation traveling wave plot 5.gif|thumb|Lin-Tsien equation traveling wave plot]]
|[[File:Lin Tsien eq TWS extended coth.gif|thumb|Lin Tsien eq TWS extended coth]]
|[[File:Lin Tsien eq TWS extended csch.gif|thumb|Lin Tsien eq TWS extended csch]]
|[[File:Lin-Tsien nlpde animation.gif|thumb|Lin-Tsien nlpde animation]]
|}
{|
|[[File:Lin-Tsien nlpde 3d plot.png|thumb|Lin-Tsien nlpde 3d plot]]
|[[File:Lin Tsien eq extended csc arctan.gif|thumb|Lin Tsien eq extended csc arctan]]
|[[File:Lin Tsien equation traveling wave plotb.gif|thumb|]]
|[[File:Lin Tsien equation traveling wave plota.gif|thumb|]]
|}

==参考文献==
<references/>
* D. Zwillinger. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p.&nbsp;138, 1997
* C. Rogers, William F. Ames, [http://books.google.gr/books?id=qyDE9V3AIn8C&dq=Lin%E2%80%93Tsien+equation+ames&source=gbs_navlinks_s Nonlinear boundary value problems in science and engineering] {{Wayback|url=http://books.google.gr/books?id=qyDE9V3AIn8C&dq=Lin%E2%80%93Tsien+equation+ames&source=gbs_navlinks_s |date=20131214074557 }}. Academic Press, p.&nbsp;373, 1989.
# *谷超豪 《[[孤立子]]理论中的[[达布变换]]及其几何应用》 上海科学技术出版社
# *阎振亚著 《复杂非线性波的构造性理论及其应用》 科学出版社 2007年
# 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
#王东明著 《消去法及其应用》 科学出版社 2002
# *何青 王丽芬编著 《[[Maple]] 教程》 科学出版社 2010 ISBN 9787030177445
#Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
# Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
#Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
#Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
#Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
#Dongming Wang, Elimination Practice,Imperial College Press 2004
# David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
# George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759
# A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108 (2018), no. 2, 359-376, https://arxiv.org/abs/1401.2122 {{Wayback|url=https://arxiv.org/abs/1401.2122 |date=20201112013741 }}
{{非线性偏微分方程}}

[[Category:微分学]]
[[Category:非线性偏微分方程]]